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Properties

Label 2-2016-7.4-c1-0-36
Degree $2$
Conductor $2016$
Sign $-0.895 + 0.444i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)5-s + 2.64·7-s + (1.32 − 2.29i)11-s − 4·13-s + (0.5 − 0.866i)17-s + (−3.96 − 6.87i)19-s + (1.32 + 2.29i)23-s + (−2 + 3.46i)25-s + 4·29-s + (−1.32 + 2.29i)31-s + (−3.96 − 6.87i)35-s + (2.5 + 4.33i)37-s − 8·41-s − 10.5·43-s + (−1.32 − 2.29i)47-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s + 0.999·7-s + (0.398 − 0.690i)11-s − 1.10·13-s + (0.121 − 0.210i)17-s + (−0.910 − 1.57i)19-s + (0.275 + 0.477i)23-s + (−0.400 + 0.692i)25-s + 0.742·29-s + (−0.237 + 0.411i)31-s + (−0.670 − 1.16i)35-s + (0.410 + 0.711i)37-s − 1.24·41-s − 1.61·43-s + (−0.192 − 0.334i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.009122700\)
\(L(\frac12)\) \(\approx\) \(1.009122700\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.32 + 2.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.96 + 6.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.32 - 2.29i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 10.5T + 43T^{2} \)
47 \( 1 + (1.32 + 2.29i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.5 + 6.06i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.32 - 2.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.32 + 2.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-4.5 + 7.79i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.32 + 2.29i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.490367825292874653841262035598, −8.411046126752311919984004429565, −7.32614737518537595684375992477, −6.59126505120635934808026150108, −5.13946341350393984729852944803, −4.93534479187327599463394063256, −4.07475957722014555136878686291, −2.83927779993662222109080614014, −1.53350288105084016813867059960, −0.35929399108056446612544159993, 1.70485322844858430998255331844, 2.63986301812814163669629027071, 3.79976341094494879058115148836, 4.47701646489852219843341050654, 5.44571403775347421711467932631, 6.58938785579002202947655824047, 7.12036056444790453987217625152, 7.944882017522494188569987200566, 8.408058216973242081617626144595, 9.661103116390698670080051268236

Graph of the $Z$-function along the critical line